As the title said, this is a Mathementation problem!

You will be given a number $$$n$$$ and you have to print two integer numbers $$$a$$$ and $$$b$$$ satisfying the following conditions: $$$$$$a^b = \prod_{d|n} d$$$$$$ $$$$$$0 \le a,b \le 10^{18}$$$$$$

$$$(\prod_{d|n} d)$$$ is the product of all divisors of $$$n$$$.

$$$\textit{Aswad}$$$ is a member of a Telegram group chat with $$$n$$$ participants, he has a unique communication style. Although $$$\textit{Aswad}$$$ he doesn't like engaging in conversations, he still like to participate in the group by responding to messages.

However, instead of responding immediately, $$$\textit{Aswad}$$$ waits for a set amount of time before sending a message. Specifically, $$$\textit{Aswad}$$$ waits for $$$k$$$ minutes after the previous message is sent before responding.

It's $$$\textbf{important}$$$ to note that if another message is sent during the waiting period, $$$\textit{Aswad}$$$ will ignore the previous message and will wait for a new $$$k$$$ minutes. This approach allows $$$\textit{Aswad}$$$ to take his time in crafting a response and to avoid getting caught up in the rapid pace of the chat.

Given a list of $$$m$$$ timestamps for when messages were sent in the group chat, and the number of minutes $$$k$$$. your task is to determine how many messages $$$\textit{Aswad}$$$ has sent in the group chat.

We all know how much it's annoying when a virus comes into your computer and start copy and paste folders until you run out of storage. However, there are antivirus which solve this problem.

But sometimes the virus comes to your computer in a human form. That's what happen to sad $$$\textit{Bashar}$$$ when he left his laptop with $$$\textit{Aswad}$$$.

To make you empathize with $$$\textit{Bashar}$$$, I will give you instructions to ruin your computer in $$$\textit{Aswad's}$$$ style and let you feel what $$$\textit{Bashar}$$$ has felt.

You are given the initial folders labeled from $$$1$$$ to $$$n$$$. For each folder you will be given the labels of folders inside it (of course there is no folder contains it self). The folder number $$$1$$$ is the Home folder.

You are given also $$$q$$$ queries. Each query consists of $$$k$$$ pairs $$$(u_i, v_i)$$$ where $$$(1 \le i \le k \le 3)$$$. And you have to do the following:

During the $$$i-th$$$ iteration $$$( 1 \leq i \leq k)$$$, You have to copy the folder $$$u_i$$$ (with all folders inside it) and paste it inside the folder $$$v_i$$$. (see the example for more clarification).

And since I want you to copy a copied folder, you have to assign labels to the newly pasted folders using the following rule: if the original folder is labeled as $$$u$$$, then the pasted node should be labeled as $$$u + n \times 2^{i-1} $$$ , where $$$i$$$ is the iteration number.

After you finish all the iterations, output the number of folders to see the terrible result.

Note that queries are independent.

On June-22-2023, Al-Baath University is supposed to hold a local contest, but the judges are still uncertain whether the problemset is easy or difficult.

The contest consists of $$$n$$$ problems, each with a value $$$a_i$$$.

One of the judges, $$$\textit{YouStill-DontKnowMeYet}$$$, suggests determining the difficulty of the contest by taking the $$$XOR$$$ value of some of the problems values and finding the largest $$$XOR$$$ value.

However, another judge, $$$\textit{Blade-Master}$$$, argues that determining the contest's difficulty is more challenging than the most challenging problem in the contest. They suggest focusing only on a range of problems from $$$l$$$ to $$$r$$$ and finding the $$$XOR$$$ value of those problems.

Later on, the judges realize that the contest is too easy, so they decide to add a problem with $$$k$$$ tags to make it more challenging.

The number of tags for a problem refers to the number of ones in its binary representation.

Given the current set of problems and the value of $$$k$$$, your task is to help the judges and determine the $$$maximum$$$ difficulty that can be achieved by adding a problem that satisfies the given conditions.

In other words, You are given an array $$$a$$$ of length $$$n$$$, you have to insert an integer $$$x$$$ $$$(0 \le x \le 1023)$$$ to the array $$$a$$$ in any position you want so the array becomes in length $$$n+1$$$.

the number $$$x$$$ should have exactly $$$k$$$ ones in the binary representation.

Your task is to find $$$\max\limits_{1 \le l \le r \le n + 1}(a_l\oplus a_{l+1} \oplus \dots \oplus a_r)$$$

Suppose you have a collection of $$$n$$$ dice, each with $$$k$$$ sides, numbered $$$1$$$ through $$$k$$$.

Or why would you suppose!!!, I will tell you what $$$\textit{Geo-Ghaffar}$$$ does in his free time and you imagine with me.

$$$\textit{Geo-Ghaffar}$$$ has a collection of $$$n$$$ dice, each with $$$k$$$ sides (numbered $$$1$$$ through $$$k$$$). When he is bored he plays the following game:

At each turn, he throws all the dice, chooses a number $$$x$$$ between $$$1$$$ and $$$k$$$ (inclusive) and discards all dice that show $$$x$$$ on their top face. He keeps doing this until He has no dice left.

Now $$$\textit{Geo-Ghaffar}$$$ is wondering that if the dice are fair, so each possible $$$\textbf{arrangement}$$$ of numbers on the dice is $$$\textbf{equally likely}$$$, What is the expected number of turns it will take to finish the game if he plays optimally. meaning he always choose the number $$$x$$$ that maximizes the number of dice that will be removed on that turn.

Since the answer may be large, print it modulo $$$998244353$$$.

Can you help him find the answer?

People think that $$$\textit{Somar}$$$ likes palindrome strings, but actually, palindrome strings mean to $$$\textit{Somar}$$$ more than his own life. That's why he always searches for them around the world.

The problem is that a palindrome string is a rare element in nature. Therefore, to obtain a palindrome string, you need to extract it from a normal string (which may or may not be a palindrome) through a difficult process. Of course, $$$\textit{Somar}$$$ is busy and needs your help to extract palindrome strings.

You will be given an array $$$w$$$ consisting of $$$n$$$ strings, each string associated with a score $$$s_i$$$.

You can perform the following operation at most $$$k$$$ times:

• Change any letter in any string to any other letter.

Your goal is to find the maximum possible score of a beautiful subarray in $$$w$$$, if you can perform the previous operation no more than $$$k$$$ times.

• We call a string array $$$a$$$ a subarray of $$$w$$$ if it can be obtained by removing any number of strings (possibly zero) from the beginning and end of $$$w$$$. Note that $$$a$$$ could be an empty array.
• We call a string array $$$a$$$ beautiful if each string in $$$a$$$ is a palindrome string.
• The score of a string array $$$a$$$ is defined as the sum of the scores of each string in $$$a$$$.
• A palindrome string is a string that remains the same when read from both ends. For example, the strings "z", "aaa", "aba", and "abccba" are palindromes, but the strings "icpc" and "ab" are not.

$$$\textit{KaitoKid}$$$ is a skilled problem solver, but he's still upset about losing the last ACPC contest to $$$\textit{UnitedBugaC}$$$. Now, he's planning to break their alliance with a secret plan that he needs your help with.

$$$\textit{KaitoKid}$$$ has $$$n$$$ friends, and he assigned a mysterious value $$$a_i$$$ to each one. He wants to take some of his friends with him, but only those who are $$$\textit{close}$$$ to each other. specifically, he wants the people who will go with him to have a degree of $$$\textit{closeness}$$$ equals to $$$x$$$.

The degree of $$$\textit{closeness}$$$ between friends is calculated as the bitwise $$$\textbf{AND}$$$ of their values. In other words, the degree of $$$\textit{closeness}$$$ between friends with values $$$b_1, b_2, \dots, b_k$$$ is: $$$b_1 \& b_2 \& \dots \& b_k$$$ where $$$\&$$$ represents the bitwise $$$\textbf{AND}$$$ operator.

$$$\textit{KaitoKid}$$$ is in a dilemma: he doesn't know how many friends he should take with him, and he's afraid to go alone. Could you help him figure out the maximum number of friends he can take with him? If no one can go with him, output $$$-1$$$.

Join $$$\textit{KaitoKid}$$$ on his mission and use your problem solving skills to save the day!

$$$\textit{ALBaath University}$$$ is hosting a competitive programming contest under the ICPC rules, and things were going great until the competitors hit some hard problems. Suddenly, they were in desperate need of some reinforcements, and some teams from other universities agreed to help them out.

But wait! Uh-oh, it turns out that outside help is strictly forbidden in this contest. $$$\textit{HIAST (Hilarious Institute for the Advanced Study of Trickery) }$$$ heard about this and sent a special team to surround the university. Its mission is to catch any team from outside that might have sneaked in.

$$$\textbf{HIAST}$$$ is tricky. It gave you the problem to solve it so It can catch your allies. Now, it's up to you if you want to sacrifice your allies just to get the problem accepted.

You are given a polygon of $$$n$$$ vertices and $$$q$$$ queries.

Each query consists of a point $$$(x,y)$$$ and you have to check if it is inside (including the borders) or outside the polygon.

The points of the polygon are given in clockwise order with the property that either $$$x_i = x_{i-1}$$$ or $$$y_i = y_{i-1}$$$ (but not both), indicating that the edges are parallel to either the $$$x$$$- or $$$y$$$-axis, Also no two edges intersect (endpoints are not considered into the intersections).

$$$\textit{Basharo}$$$ is too much into counting problems and he challenge you to solve this one.

You'll be given a connected undirected graph of $$$n$$$ vertices and $$$n-1$$$ edges. The vertices are numbered from $$$1$$$ to $$$n$$$ and the vertex with the number $$$i$$$ is colored with the color $$$c_i$$$. The edges are also numbered from $$$1$$$ to $$$n-1$$$.

The path from a vertex $$$u$$$ to a vertex $$$v$$$ is called beautiful if it satisfies these conditions:

• $$$u \lt v$$$
• $$$gcd(c_u,c_v) = 1$$$. Where $$$gcd(x,y)$$$ is the greatest common divisor of the numbers $$$x$$$ and $$$y$$$.

Your task is to count for each edge $$$i$$$ the number of beautiful paths that contain this edge.

In a parallel universe, a contest has been organized by advanced problem setters, and $$$\textit{Kifah}$$$ is one of them. The chief judge asked $$$\textit{Kifah}$$$ to set an extreme problem. Unfortunately, $$$\textit{Kifah}$$$ didn't understand the meaning of "extreme problem" and was too shy to ask for clarification. It's possible that he understood "extreme" in the opposite sense. As the problem is being held in a far universe, no one can catch $$$\textit{Kifah}$$$, and the contestants cannot seek revenge.

Unfortunately for you, there is a bug in the universe's system which directed the problem to your contest. I'm sorry for that, and I hope you enjoy your hard luck.

In this Problem you will be given an array $$$a_1,a_2,...,a_n$$$.

You can do the following operation any number of times. During each operation you can choose one element $$$a_i$$$ from the array and do one of the following two actions:

• Choose one bit from the binary representation of $$$a_i$$$ and turn it off.
• Choose two bits from the binary representation of $$$a_i$$$ and turn them both on (the two chosen bits have to be both zeros before turning them on)

What is the minimum number of operations you have to do to make all elements of array $$$a$$$ equal.

On a boring Friday night, $$$\textit{Ramez}$$$ and his son $$$\textit{Molham}$$$ were sitting together. As usual every Friday, each Dad in the world should ask his son to do something. So, $$$\textit{Ramez}$$$ asked his son to solve the following problem:

He gave $$$\textit{Molham}$$$ $$$n$$$ cards, numbered from 1 to $$$n$$$ and arranged in a clockwise circle shape.

Initially, all the cards are black. And $$$\textit{Ramez}$$$ asked $$$\textit{Molham}$$$ to paint $$$n-1$$$ cards with red (if it is possible).

$$$\textit{Molham}$$$ can make the painting operation as follows:

He can paint a black card ($$$i$$$) with red if and only if there is another black card ($$$j$$$) and there is a way to get from $$$i$$$ to $$$j$$$ passing through exactly two cards differ than $$$i$$$ and $$$j$$$ .

$$$\textit{Molham}$$$ thought the problem is so easy, but $$$\textit{Ramez}$$$ did not stop here; if there are many possible orders to paint the $$$n-1$$$ cards red, he asked for the minimum lexicographically order of painting.

The coloring order is defined as follows:

If at the $$$k$$$-th operation $$$\textit{Molham}$$$ colored the card with number $$$i$$$ with red (if it is possible), then the $$$k$$$-th number of the coloring order should be $$$i$$$.

After this extra condition, $$$\textit{Molham}$$$ found the problem so hard so he asked for your help.

We all dreamed about that teacher who gives us the mark we want. Lately we started dreaming about the teacher who gives as what we deserve at least.

Today is your day, we are giving you the chance to write a code which take the mark of the student and change it to $$$10$$$ directly then print it.

But wait, we can't give a student with an empty paper a mark of $$$10$$$. so at least their origin mark should be greater than zero to get $$$10$$$.

You are in front of a hard grid challenge.

There are a grid, a clock, and of course a coin on the grid. The clock starts counting from the moment number 1, then number 2, and so on. At Each moment you can move the coin according to the following rule:

At each moment $$$i$$$ you can choose an integer number $$$x_i$$$. Let's assume that the coin is at the coordinate $$$(a,b)$$$ at the $$$ith$$$ moment. Now if $$$i$$$ is odd, the coin will jump to the coordinate $$$(a+x_i,b+x_i)$$$. But if $$$i$$$ is even, the coin will jump to the coordinate $$$(a+x_i,b-x_i)$$$.

The challenge is to find the minimum number of moments needed to move the coin from the coordinate $$$(0,0)$$$ to the coordinate $$$(n,m)$$$ and what is the sequence $$$x_i$$$ you have chosen to solve the challenge.

The challenge may be Impossible, so in this case print -1.